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[Community Question] Linear-algebra: Explicit example of an additive map which is not R-linear

One of our user asked:

Is there an explicit example of an additive map $\mathbb{R}^n \rightarrow \mathbb{R}^m$ which is not linear? (I have mostly thought about the question when $m = n = 1$, and I don't think the general case is any easier.) I know that something like $f: \mathbb{C} \rightarrow \mathbb{C}$ which sends $f: z \mapsto \text{Real}(z)$ would be additive but not $\mathbb{C}$-linear. I also know that since $\mathbb{R}$ is a $\mathbb{Q}$-vector space, I can find some example where $1 \mapsto 1$ and $\sqrt{2} \mapsto 0$. Is there an explicit example? By explicit, I mean, given an element in the domain, there would be some procedure to decide where it maps. Thank you!


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